Universality of Dicke superradiance in arrays of quantum emitters

Dicke superradiance is an example of emergence of macroscopic quantum coherence via correlated dissipation. Starting from an initially incoherent state, a collection of excited atoms synchronizes as they decay, generating a macroscopic dipole moment and emitting a short and intense pulse of light. While well understood in cavities, superradiance remains an open problem in extended systems due to the exponential growth of complexity with atom number. Here we show that Dicke superradiance is a universal phenomenon in ordered arrays. We present a theoretical framework – which circumvents the exponential complexity of the problem – that allows us to predict the critical distance beyond which Dicke superradiance disappears. This critical distance is highly dependent on the dimensionality and atom number. Our predictions can be tested in state of the art experiments with arrays of neutral atoms, molecules, and solid-state emitters and pave the way towards understanding the role of many-body decay in quantum simulation, metrology, and lasing.


I. DECAY RATES AS A FUNCTION OF DISTANCE IN ORDERED ARRAYS
Generally, the variance of the eigenvalues {Γ ν } increases with decreasing inter-atomic distances. However, this is not always strictly true. At some specific distances, there are geometric resonances that cause the decay rates to experience sudden changes [1][2][3][4], which leads to an increase in the variance, as shown in Supplementary Figure 1. These resonances are associated with far-field contributions to the interaction, and occur because certain decay channels become significantly brighter due to constructive interference. In 1D, the first revival occurs at d = λ 0 /2. Extremely subradiant states do not exist for this distance, and thus this revival is not enough to enhance two-photon emission and superradiance. In 2D, for atoms polarized perpendicular to the surface, revivals occur at d = λ 0 /2 and d = λ 0 / √ 2. For these distances in 2D there are subradiant states. The revivals are strong enough to cause superradiance, leading to the non monotonic behavior of the critical distance with atom number observed in Fig. 4(b) in the main text. For atoms with polarization in the plane, far-field emission in the plane is forbidden in the direction that coincides with that of the polarization, greatly quenching the revivals.

II. ROLE OF HAMILTONIAN INTERACTIONS IN DEPHASING
We consider the role of the Hamiltonian by considering a delay time between the first two photons and comparing to the case without a delay. We calculate on the fully excited state. This is shown in Supp. Figure 2(a) at the critical distance for different arrays. We note that the Hamiltonian causes very slow dephasing in the case of a linear or square array, and has no impact on the ring array. Calculations show that mixing due to non-measurement introduces an additional (but smaller) dephasing.
The dephasing is reduced with N , as shown in Supplementary Figure 2(b) at the critical distance. Hamiltonian dephasing is primarily due to inhomogeneous (i.e., local) frequency shifts caused by interactions [5]. With increasing N , d critical increases, such that interactions are reduced at the critical distance and dephasing is reduced. Furthermore, atoms that see the similar local environment have similar shifts. This means that the inhomogeneity reduces as N increases, as the fraction of atoms in the "bulk" vs the edges increases with N . This effect is more pronounced for the chain, where the fraction of bulk atoms scales as 1/N , than the square array, where the fraction scales as 1/ √ N , as can be seen in the inset to Supplementary Figure 2

III. DERIVATION OF g (2) (0) FOR AN IMPERFECTLY PREPARED INITIAL STATE
Here we consider the role of "single-hole" imperfections, i.e., where not all atoms are in the excited state. This state reads where ζ a is the complex coefficient for the single-hole state in which atom a is in the ground state.
The quantities required to calculate g (2) (0) do not mix states with different excitation numbers so we can evaluate the single-hole contribution separately to the fully-excited contribution. On the single-hole state, the expectation values required to calculate g (2) (0) are calculated as We calculate the numerator and denominator of g (2) (0) separately. The denominator is as follows Following a similar procedure, the numerator is readily found to be N ν,µ=1 We can now combine these with the fully-excited results to find g (2) (0) for the state given by Eq. (2) g (2) To investigate the impact of the imperfect initial state, we consider coherent spin states of the form These would be produced experimentally by a short, intense pulse of duration τ {(N Γ 0 ) −1 , J −1 12 }. Here, we consider ϕ ≈ 1 such that we truncate the state to the form (here left unnormalized for simplicity) and use Eq. (6) to calculate the critical distance for imperfect initial states. Supplementary Figure 3(a) shows that the impact is marginal. For a total imperfection of 15%, the critical distance drops by a factor of only 0.4%. For these small imperfections in the initial state, the relative decrease in d critical is approximately linear, and seems to be independent of the array geometry.

IV. CRITICAL DISTANCE IN THE PRESENCE OF CLASSICAL SPATIAL DISORDER
Supplementary Figure 3(b) shows that superradiance is robust to classical disorder the position of in the emitters. We add a randomly-generated 3D Gaussian noise to each emitter position with standard deviation σ r in all directions. We stochastically generate a large number of arrays and find the critical distance at which superradiance is lost.

V. CONSIDERATIONS FOR SOLID-STATE EMITTERS
Solid-state emitters constitute an alternative platform for producing emitter arrays, as strongly sub-wavelength distances can be achieved simply through fabrication, without the need for optical trapping. However, these emitters have other issues that may negatively impact collective decay. Here, we consider the impact of inhomogeneous broadening and non-radiative decay.

A. Inhomogeneous broadening
For non-identical emitters, we define each emitter to have frequency ω i 0 and spontaneous emission rate Γ i 0 , with mean valuesω 0 andΓ 0 . If the frequency broadening is small, such that the spectral response is flat across the range of ω i 0 , then frequency broadening does not impact the treatment of the dissipation and we can follow the derivation of g (2) (0) above with the alterations that the operator decay rates now obey Therefore g (2) This can be recast in terms of two variances as This expression is maximized for zero inhomogeneity, i.e. Γ i 0 = Γ 0 , and so inhomogeneous broadening in the emitter decay rates strictly increases dephasing.

B. Non-radiative decay
Solid-state emitters can decay without emitting light. We consider that this type of decay is not correlated (i.e., it is local). The master equation thus readṡ where γ i is the non-radiative decay rate of atom i. We then write g (2) (0) as where p(0, j) is the probability of zero non-radiative events before the emission of j photons, and p i (l, m) is the probability of a single non-radiative event occurring on atom i right before the mth photon during the emission of l photons. Terms with two or more non-radiative events are assumed to be negligible and are hence ignored, as we assume the non-radiative decay to be small, γ i Γ 0 . We wish to expand g (2) (0) in the same manner as above, which requires the evaluation of the expectation values By noting that Eqs. (14b) and (14c) yield the same result, and substituting in the expressions for terms without non-radiative terms from above, we arrive to We are interested in calculating g (2) (0) around the critical distance, where the second photon is emitted at approximately the same rate as the first, N Γ 0 . In this situation, we can approximate the probabilities as whereγ is the mean non-radiative decay rate. This approximation should also be valid for large N , where the emission of the first photon does not substantially alter the rate of the second photon. This simplifies the expression to We thus need to calculate and N i,ν,µ=1 Combining these two expressions we obtain the second order correlation function near the critical distance as g (2) (0) = (20) If each emitter has the same non-radiative decay rate γ, this simplifies to (21)

C. Superradiance with solid-state emitters
Superradiance persists in the presence of non-radiative decay and inhomogeneous broadening. Supplementary  Figure 4(a) shows that the superradiant burst survives levels of non-radiative decay as large as those of radiative decay. Nevertheless, increased non-radiative decay rates enhance dephasing, eventually destroying superradiance as the emission pathways are dominated by non-radiative routes. As a result, the critical distance at which the superradiant burst disappears is shifted to smaller distances. Supplementary Figure 4(b) shows that the superradiant burst survives inhomogeneous broadening on the emitter resonance frequency even at levels beyond 10 times the Supplementary Figure 4. Impact of non-radiative decay and inhomogeneous broadening on superradiance. (a,b) Photon emission rate from an initially inverted square array of 3 × 3 emitters and inter-atomic spacing d = 0.1λ0 in the presence of (a) non-radiative decay and (b) inhomogeneous broadening on the emitter resonance frequencies. In (b), plotted curves are the average of 100 stochastically generated instances with Gaussian distributed noise of width σω. (c) Boundaries between the burst (colored) and no-burst (white) regions as a function of inter-particle distance d and emitter number for square arrays with and without non-radiative decay. The symbols and represent points where, with decreasing d, g (2) (0) goes above and below unity, respectively. (d) Critical distance for square arrays of 8 × 8 emitters as a function of non-radiative decay rate. In the presence of inhomogeneous broadening, the decay rate of each emitter is calculated as a random sample of a Gaussian distribution with mean Γ0 and standard deviation σ Γ 0 . Circles represent individual stochastic samples, and the solid line shows the average of 100 samples. In all cases, emitters are polarized perpendicular to the array and are assumed to have the same non-radiative decay rate γ.